ii) Estimators of the percentiles are not uniquely defined, but some are more optimal than others in terms of bias and variability. For example, suppose we want to estimate the median. We could average the three middle observations or just use the intro to stat definition and use the middle observation as the estimator of the median. If the population is symmetric both estimators will be unbiased estimators of the population median, yet by using the three middle observations we would reduce the variablility of the estimator of the median substantially.
I'm not sure I understand the reason that the average of the three middle observations would be a "less" biased estimate of the median than the middle observation by itself. If we extend the number of "center" values averaged to higher levels (center 5 or 7 or 333) we would be moveing closer to the mean (and thus away from the median). Perhaps there is some probability idea at work that I don't see or understand. Could Alan or someone please explain.
Pat Ballew Misawa, Japan
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